Nondense orbits on homogeneous spaces and applications to geometry and number theory
2020
Let $G$ be a Lie group, $\Gamma\subset G$ a discrete subgroup, $X=G/\Gamma$, and $f$ an affine map from $X$ to itself. We give conditions on a submanifold $Z$ of $X$ guaranteeing that the set of points $x\in X$ with $f$-trajectories avoiding $Z$ is hyperplane absolute winning (a property which implies full \hd\ and %stability {is stable} under countable intersections). A similar result is proved for one-parameter actions on $X$. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.
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